This is my favorite strategy, because it always makes me feel amazing when I spot it!
I'm hoping to explain the blind dot rule by taking you along a similar journey I took to discover the blind dot rule. Enjoy :)
In a time before I came to fully understand what the blind dot rule was, I started to pick up on these areas that looked like little crevices (this is what we will call them for now). There was some sort of pattern going on, but I couldn't put my finger on what it was.
Here's a 5x5 board I got one day, full of these crevices:
⚫️2️⃣🔴⚫️🔴
⚫️⚫️4️⃣⚫️⚫️
⚫️⚫️3️⃣⚫️⚫️
3️⃣🔴3️⃣⚫️⚫️
🔴⚫️⚫️🔴⚫️
Let's solve step-by-step, and listen to what these crevices (denoted by 🔳) tell us:
⚫️2️⃣🔴🔳🔴
⚫️⚫️4️⃣⚫️⚫️
⚫️⚫️3️⃣⚫️⚫️
3️⃣🔴3️⃣⚫️⚫️
🔴🔳⚫️🔴🔳
🔵2️⃣🔴⚫️🔴
🔵⚫️4️⃣⚫️⚫️
🔵⚫️3️⃣⚫️⚫️
3️⃣🔴3️⃣⚫️⚫️
🔴⚫️⚫️🔴⚫️
🔵2️⃣🔴⚫️🔴
🔵🔵4️⃣⚫️⚫️
🔵🔴3️⃣⚫️⚫️
3️⃣🔴3️⃣⚫️⚫️
🔴⚫️⚫️🔴⚫️
The 4️⃣ can see all its dots.
🔵2️⃣🔴⚫️🔴
🔵🔵4️⃣🔴⚫️
🔵🔴3️⃣⚫️⚫️
3️⃣🔴3️⃣⚫️⚫️
🔴⚫️🔴🔴⚫️
🔵2️⃣🔴🔴🔴
🔵🔵4️⃣🔴⚫️
🔵🔴3️⃣⚫️⚫️
3️⃣🔴3️⃣⚫️⚫️
🔴🔴🔴🔴⚫️
Alright, pause.
It seems that those 2 crevices became red! There was also a new crevice made in the process. Maybe those will turn red too?
🔵2️⃣🔴🔴🔴
🔵🔵4️⃣🔴🔳
🔵🔴3️⃣⚫️⚫️
3️⃣🔴3️⃣⚫️⚫️
🔴🔴🔴🔴🔳
Let's continue solving:
🔵2️⃣🔴🔴🔴
🔵🔵4️⃣🔴⚫️
🔵🔴3️⃣🔵⚫️
3️⃣🔴3️⃣🔵⚫️
🔴🔴🔴🔴⚫️
Both 3️⃣s can see all their dots.
🔵2️⃣🔴🔴🔴
🔵🔵4️⃣🔴⚫️
🔵🔴3️⃣🔵🔴
3️⃣🔴3️⃣🔵🔴
🔴🔴🔴🔴⚫️
🔵2️⃣🔴🔴🔴
🔵🔵4️⃣🔴🔴
🔵🔴3️⃣🔵🔴
3️⃣🔴3️⃣🔵🔴
🔴🔴🔴🔴🔴
If you paid especially close attention, you might've noticed that not only did all the crevices become red, every "entrance" to their cove became red too.
One explanation involves the property explained in a previous post, that a board has one, and only one solution.
Suppose we had a board like this instead:
🔵2️⃣🔴🔴🔴
🔵🔵⚫️🔴⚫️
🔵🔴3️⃣🔵⚫️
3️⃣🔴3️⃣🔵⚫️
🔴🔴🔴🔴⚫️
Let's assume that they are blues:
🔵2️⃣🔴🔴🔴
🔵🔵⚫️🔴⚫️
🔵🔴3️⃣🔵🔵
3️⃣🔴3️⃣🔵🔵
🔴🔴🔴🔴⚫️
Then the crevices would be ambiguous, or allow the board to contain more than 1 solution:
🔵2️⃣🔴🔴🔴
🔵🔵🔴🔴❔
🔵🔴3️⃣🔵🔵
3️⃣🔴3️⃣🔵🔵
🔴🔴🔴🔴❔
So, if it can't be blue, then it must be red!
🔵2️⃣🔴🔴🔴
🔵🔵🔵🔴⚫️
🔵🔴3️⃣🔵🔴
3️⃣🔴3️⃣🔵🔴
🔴🔴🔴🔴⚫️
Let's use our new understanding to find and solve other boards using this type of pattern.
...hey look!
5️⃣⚫️⚫️5️⃣⚫️
⚫️⚫️4️⃣⚫️3️⃣
⚫️⚫️⚫️⚫️3️⃣
⚫️⚫️⚫️🔴⚫️
⚫️🔴🔴🔴🔴
What about this one? Something seems special around here:
5️⃣⚫️⚫️5️⃣⚫️
⚫️⚫️4️⃣⚫️3️⃣
⚫️⚫️⚫️⚫️3️⃣
⚫️🔳⚫️🔴⚫️
⚫️🔴🔴🔴🔴
If you imagine each blue dot is like a 4-way laser, it would cover the entire board except for that spot!
5️⃣⚫️⚫️5️⃣⚫️
⬇️⚫️4️⃣⚫️3️⃣
⬅️⬅️⬅️⬅️3️⃣
⬇️😎⬇️🔴⚫️
⬇️🔴🔴🔴🔴
In a way, it's kind of like the crevices. They also don't have any blue dots looking at them. See?
⚫️2️⃣🔴😎🔴
⚫️⚫️4️⃣➡️➡️
⚫️⚫️3️⃣⚫️⚫️
3️⃣🔴3️⃣➡️➡️
🔴😎⬇️🔴😎
Let's solve it and see what happens. Make a prediction!
Click to show solving steps
5️⃣⚫️⚫️5️⃣⚫️
⚫️⚫️4️⃣🔵3️⃣
⚫️⚫️⚫️⚫️3️⃣
⚫️⚫️⚫️🔴⚫️
⚫️🔴🔴🔴🔴
5️⃣⚫️⚫️5️⃣🔴
⚫️🔴4️⃣🔵3️⃣
⚫️⚫️⚫️⚫️3️⃣
⚫️⚫️⚫️🔴🔴
⚫️🔴🔴🔴🔴
5️⃣⚫️⚫️5️⃣🔴
⚫️🔴4️⃣🔵3️⃣
⚫️⚫️🔵🔵3️⃣
⚫️⚫️⚫️🔴🔴
⚫️🔴🔴🔴🔴
5️⃣⚫️⚫️5️⃣🔴
⚫️🔴4️⃣🔵3️⃣
⚫️🔴🔵🔵3️⃣
⚫️⚫️⚫️🔴🔴
⚫️🔴🔴🔴🔴
5️⃣🔵🔵5️⃣🔴
⚫️🔴4️⃣🔵3️⃣
⚫️🔴🔵🔵3️⃣
⚫️⚫️⚫️🔴🔴
⚫️🔴🔴🔴🔴
5️⃣🔵🔵5️⃣🔴
🔵🔴4️⃣🔵3️⃣
🔵🔴🔵🔵3️⃣
⚫️⚫️⚫️🔴🔴
⚫️🔴🔴🔴🔴
5️⃣🔵🔵5️⃣🔴
🔵🔴4️⃣🔵3️⃣
🔵🔴🔵🔵3️⃣
🔴⚫️⚫️🔴🔴
🔴🔴🔴🔴🔴
5️⃣🔵🔵5️⃣🔴
🔵🔴4️⃣🔵3️⃣
🔵🔴🔵🔵3️⃣
🔴⚫️🔴🔴🔴
🔴🔴🔴🔴🔴
5️⃣🔵🔵5️⃣🔴
🔵🔴4️⃣🔵3️⃣
🔵🔴🔵🔵3️⃣
🔴🔴🔴🔴🔴
🔴🔴🔴🔴🔴
Cool! The dot indeed ended up red! Not only that, but ALL 3 of its neighbors ended up red as well!
It seems like the pattern has less to do with crevices, and more to do with whether any numbered blue dot can see a given dot.
Let's call the dots that can't be seen blind dots. For the dots immediately adjacent to the blind dot, let's call them guard dots.
Let's go over one final example for this post. In the previous examples, we assumed the line of sight of numbered blue dots extends arbitrarily outward, but I want to show that there is technically a limit.
Let's demonstrate by example, using this board:
🔴⚫️⚫️⚫️1️⃣
🔴🔴2️⃣⚫️⚫️
🔴1️⃣⚫️⚫️2️⃣
⚫️⚫️⚫️2️⃣⚫️
🔴⚫️⚫️3️⃣⚫️
Here are the areas I'm claiming to be blind dots:
🔴🔳⚫️⚫️1️⃣
🔴🔴2️⃣⚫️⚫️
🔴1️⃣⚫️⚫️2️⃣
🔳⚫️⚫️2️⃣⚫️
🔴⚫️⚫️3️⃣⚫️
You might think these can't be blind dots, because the numbered dots has line of sight to them:
🔴😧⬅️⬅️#️⃣
🔴🔴⚫️⚫️⚫️
🔴⚫️⚫️⚫️⚫️
😧⬅️⬅️#️⃣⚫️
🔴⚫️⚫️⚫️⚫️
But they don't have line of sight, because the 1️⃣ in the top right can only see one dot to the west, and likewise, the 2️⃣ near the bottom can only see one more dot to the west before it would see all of its dots.
🔴😎❌🔵1️⃣
🔴🔴⚫️⚫️⚫️
🔴⚫️⚫️⚫️⚫️
😎❌🔵2️⃣⚫️
🔴⚫️⚫️3️⃣⚫️
Of course, let's solve it to double check!
Solution steps
🔴⚫️⚫️⚫️1️⃣
🔴🔴2️⃣⚫️⚫️
🔴1️⃣⚫️⚫️2️⃣
⚫️⚫️⚫️2️⃣⚫️
🔴⚫️⚫️3️⃣⚫️
Looking further south would exceed the 1️⃣:
🔴⚫️🔴🔵1️⃣
🔴🔴2️⃣⚫️🔴
🔴1️⃣⚫️⚫️2️⃣
⚫️⚫️⚫️2️⃣⚫️
🔴⚫️⚫️3️⃣⚫️
One specific dot included in all solutions imaginable for the 2️⃣:
🔴⚫️🔴🔵1️⃣
🔴🔴2️⃣⚫️🔴
🔴1️⃣🔵⚫️2️⃣
⚫️⚫️⚫️2️⃣⚫️
🔴⚫️⚫️3️⃣⚫️
The 1️⃣ can see all its dots:
🔴⚫️🔴🔵1️⃣
🔴🔴2️⃣⚫️🔴
🔴1️⃣🔵🔴2️⃣
⚫️🔴⚫️2️⃣⚫️
🔴⚫️⚫️3️⃣⚫️
Only one direction left to look in for the 2️⃣:
🔴⚫️🔴🔵1️⃣
🔴🔴2️⃣⚫️🔴
🔴1️⃣🔵🔴2️⃣
⚫️🔴⚫️2️⃣🔵
🔴⚫️⚫️3️⃣🔵
The 2️⃣ near the bottom can see all its dots:
🔴⚫️🔴🔵1️⃣
🔴🔴2️⃣⚫️🔴
🔴1️⃣🔵🔴2️⃣
⚫️🔴🔴2️⃣🔵
🔴⚫️⚫️3️⃣🔵
Only one direction left to look in for the 2️⃣ and 3️⃣:
🔴⚫️🔴🔵1️⃣
🔴🔴2️⃣🔵🔴
🔴1️⃣🔵🔴2️⃣
⚫️🔴🔴2️⃣🔵
🔴⚫️🔵3️⃣🔵
The 3️⃣ can see all its dots:
🔴⚫️🔴🔵1️⃣
🔴🔴2️⃣🔵🔴
🔴1️⃣🔵🔴2️⃣
⚫️🔴🔴2️⃣🔵
🔴🔴🔵3️⃣🔵
Yep! Those are blind dots!
🔴🔴🔴🔵1️⃣
🔴🔴2️⃣🔵🔴
🔴1️⃣🔵🔴2️⃣
🔴🔴🔴2️⃣🔵
🔴🔴🔵3️⃣🔵
In summary, we learned two things:
- A blind dot is a dot that cannot be reached or seen by a blue dot with a number
- A blind dot and its 4 adjacent guard dots are to be filled in with red walls
The blind dot rule is helpful because it can place red dots in a way that constrains numbered blue dots, which makes it quicker to count.
In the next part, I'll explain the deal with the guard dots. It turns out, there is more to the blind dot rule than meets the eye ;)
See you in the next one!